One of the easiest examples to understand is the simple car,
which is shown in Figure 13.1. We all know that a car
cannot drive sideways because the back wheels would have to slide
instead of roll. This is why parallel parking is challenging. If all
four wheels could be turned simultaneously toward the curb, it would
be trivial to park a car. The complicated maneuvers for parking a
simple car arise because of rolling constraints.

Figure 13.1:
The simple car has three degrees of
freedom, but the velocity space at any configuration is only
two-dimensional.

The car can be imagined as a rigid body that moves in the plane.
Therefore, its C-space is
. Figure
13.1 indicates several parameters associated with the
car. A configuration is denoted by
. The body
frame of the car places the origin at the center of rear axle, and the
-axis points along the main axis of the car. Let denote the
(signed) speed13.2 of the car. Let denote the
steering angle (it is negative for the wheel orientations shown in
Figure 13.1). The distance between the front and rear
axles is represented as . If the steering angle is fixed at
, the car travels in a circular motion, in which the radius of
the circle is . Note that can be determined from the
intersection of the two axes shown in Figure 13.1 (the
angle between these axes is ).

Using the current notation, the task is to represent the motion of the
car as a set of equations of the form

(13.11)

In a small time interval, , the car must move approximately
in the direction that the rear wheels are pointing. In the limit as
tends to zero, this implies that
.
Since
and
, this condition can be written as a Pfaffian
constraint (recall (13.5)):

(13.12)

The constraint is satisfied if
and
. Furthermore, any scalar multiple of this solution is also a
solution; the scaling factor corresponds directly to the speed
of the car. Thus, the first two scalar components of the
configuration transition equation are
and
.

The next task is to derive the equation for
. Let
denote the distance traveled by the car (the integral of speed). As
shown in Figure 13.1, represents the radius of a
circle that is traversed by the center of the rear axle, if the
steering angle is fixed. Note that
. From
trigonometry,
, which implies

(13.13)

Dividing both sides by and using the fact that
yields

(13.14)

So far, the motion of the car has been modeled, but no action
variables have been specified. Suppose that the speed and
steering angle are directly specified by the action variables
and , respectively. The convention of using a
variable with the old variable name appearing as a subscript will be
followed. This makes it easy to identify the actions in a
configuration transition equation. A two-dimensional action vector,
, is obtained. The configuration transition
equation for the simple car is

(13.15)

As expressed in (13.15), the transition equation is not yet
complete without specifying , the set of actions of the form
. First suppose that any
is possible.
What steering angles are possible? The interval
is
sufficiently large for the steering angle because any other
value is equivalent to one between and . Steering
angles of and are problematic. To derive the
expressions for and , it was assumed that the car moves
in the direction that the rear wheels are pointing. Imagine you are
sitting on a tricycle and turn the front wheel perpendicular to the
rear wheels (assigning
). If you are able to pedal,
then the tricycle should rotate in place. This means that
because the center of the rear axle does not translate.

This strange behavior is not allowed for a standard automobile. A car
with rear-wheel drive would probably skid the front wheels across the
pavement. If a car with front-wheel drive attempted this, it should
behave as a tricycle; however, this is usually not possible because
the front wheels would collide with the front axle when turned to
. Therefore, the simple car should have a maximum
steering angle,
, and we require that
. Observe from Figure 13.1 that a maximum
steering angle implies a minimum turning radius,
.
For the case of a tricycle,
. You may have
encountered the problem of a minimum turning radius while trying to
make an illegal U-turn. It is sometimes difficult to turn a car
around without driving it off of the road.

Now return to the speed . On level pavement, a real vehicle has
a top speed, and its behavior should change dramatically depending on
the speed. For example, if you want to drive along the minimum
turning radius, you should not drive at 140km/hr. It seems that the
maximum steering angle should reduce at higher speeds. This enters
the realm of dynamics, which will be allowed after phase spaces are
introduced in Section 13.2. Following this, some models of
cars with dynamics will be covered in Sections 13.2.4 and
13.3.3.

It has been assumed implicitly that the simple car is moving slowly to
safely neglect dynamics. A bound such as
can be placed
on the speed without affecting the configurations that it can reach.
The speed can even be constrained as
without
destroying reachability. Be careful, however, about a bound such as
. In this case, the car cannot drive in reverse!
This clearly affects the set of reachable configurations. Imagine a
car that is facing a wall and is unable to move in reverse. It may be
forced to hit the wall as it moves.

Based on these considerations regarding the speed and steering angle,
several interesting variations are possible:

[] Tricycle:. Assuming front-wheel drive, the ``car'' can rotate
in place if
or
. This is unrealistic
for a simple car. The resulting model is similar to that of the
simple unicycle, which appears later in (13.18).

[] Simple Car [596]:. By requiring that
, a car with minimum turning radius
is obtained.

[] Reeds-Shepp Car
[814,923]: Further restrict the
speed of the simple car so that
.13.3 This model intuitively
makes correspond to three discrete ``gears'': reverse, park, or
forward. An interesting question under this model is: What is the
shortest possible path (traversed in
by the center of the rear
axle) between two configurations in the absence of obstacles? This is
answered in Section 15.3.

[] Dubins Car [294]:
Remove the reverse speed from the
Reeds-Shepp car to obtain
as the only possible
speeds. The shortest paths in
for this car are quite
different than for the Reeds-Shepp car; see Section
15.3.

The car that was shown in Figure 1.12a of Section
1.2 is even more restricted than the Dubins car because it
is additionally forced to turn left.

Basic controllability issues have been studied thoroughly for the
simple car. These will be covered in Section 15.4, but it
is helpful to develop intuitive notions here to assist in
understanding the planning algorithms of Chapter 14. The
simple car is considered nonholonomic because there are
differential constraints that cannot be completely integrated. This
means that the car configurations are not restricted to a lower
dimensional subspace of . The Reeds-Shepp car can be maneuvered
into an arbitrarily small parking space, provided that a small amount
of clearance exists. This property is called small-time local
controllability and is presented in Section
15.1.3. The Dubins car is nonholonomic, but it
does not possess this property. Imagine the difficulty of parallel
parking without using the reverse gear. In an
infinitely large parking lot without obstacles, however, the Dubins
car can reach any configuration.